Chapter 9, Problem 1E

Let $H=\left\{\left(1\right),\left(12\right)\right\}$ . Is H normal in $S_3$ ?

To determine

To check whether $H$ is normal or not in $S_3$ .

Answer to Problem 1E

$H$ is not normal in $S_3$

Explanation of Solution

Given:

The subgroup is $H=\{(1),(12)$

Calculation:

Consider the following result related to normal subgroups:

A subgroup $H$ of a group $G$ is normal if and only if

$ah{a}^{-1}\in Hforeverya\in Gandh\in H$ $aH=Haforalla\in G$ .....(1)

Another condition for subgroup to be normal is

$aH{a}^{-1}\subseteq Hforalla\in G$ ..... (2)

That is

Use the result (1) to check whether $H$ is normal or not in $H$ as follows:

Take $a=(13)$ and check the condition as follows:

$(13)H=\{(13),(13)(12)\}=\{(13),(123)\}$

On the other hand compute $H(13)$

$H(13)=\{(13),(12)(13)\}=\{(13),(132)\}$

Since $(13)H\ne H(13)$ , this contradicts the result (1).

So $H$ is not normal in $S_3$